Abstract
In this paper, we give new conditions for existence and uniqueness of a best proximity point for Geraghty- and Caristi-type mappings. The presented results are most valuable generalizations of the Geraghty and Caristi fixed point theorems.
1. Introduction and Preliminaries
The Banach contraction principle (BCP) in metric spaces has been generalized and extended in various ways. As a generalization of the BCP, Geraghty [1] proved the following.
Theorem 1 [1]. Let be a complete metric space and let be an operator. Suppose that there exists satisfying the condition If satisfies the following inequality then has a unique fixed point.
The following are two examples of Geraghty functions .
One of the important extensions of the BCP was given by Caristi [2].
Theorem 2 [2]. Let be a self-mapping on a complete metric space . Assume that there is a bounded below and lower semicontinuous function so that for all . Then, possesses a fixed point.
On the other hand, Kirk et al. [3] in 2003 introduced the notion of a cyclic representation.
Definition 3 [3]. Let and be nonempty subsets of a metric space and . Then, is called a cyclic map if and .
The following interesting theorem for a cyclic map was given in [3].
Theorem 4 [3]. Let and be nonempty closed subsets of a complete metric space . Suppose that is a cyclic map such that for all and , where is a constant. Then, has a unique fixed point and .
Let and be nonempty sets of a metric space . Given a map, . An element is called a best proximity point of if
The set of best proximity points of is denoted by . The research of best proximity points is meaningful in optimization. The problem of existence of best proximity points in uniformly convex Banach spaces and in metric spaces as well as the convergence of sequences to such points has been focused on and successfully solved in some classic pioneering works (see [4]).
Definition 5 [5]. Let be a metric space and and be subsets of . A map is said to be a cyclic contractive map if it satisfies (i), for all and .
Eldred and Veeramani [5] extended Theorem 4 to include the case , by the following existence result of a best proximity point.
Theorem 6 [5]. Let and be nonempty closed subsets of a metric space and let be a cyclic contraction map. If either or is boundedly compact, then there exists such that
A convenience attention has been recently devoted to the research on existence and uniqueness of best proximity points of self-mappings, as well as, to the investigation of associated relevant properties, for instance, stability of the iterations. The various related performed researches include the cases of cyclic -contractions [6, 7], cyclic Meir-Keeler contractions [8], weak cyclic Bianchini contractions [9], weak cyclic Kannan contractions [10], -cyclic summing iterated contractions [11], and MF-cyclic contractions with Property UC [12]. Some contractive conditions and related properties under general contractive conditions including some proximal rational types have been also investigated [13]. In this paper, we ensure the existence of best proximity points for Geraghty and Caristi type contraction mappings.
2. A Best Proximity Point Result for Geraghty-Type Contractions
In this section, we introduce cyclic Geraghty contraction maps and give new conditions for existence and uniqueness of a best proximity point.
Definition 7. Let be a complete metric space and and be subsets of . A map is a cyclic Geraghty contraction map if there exists such that (i) for all where is the set of functions so thatif and , then ;if and , then .
We give the following theorem (comparable to Theorem 3.1 of [1]).
Theorem 8. Let and be closed subsets of a complete metric space such that . Suppose is a cyclic Geraghty contraction map. Then, . Further, if and , then converges to a best proximity point.
Proof. Fix and define a sequence in by , . First, we show that We have
Hence, is monotonic decreasing and bounded below. So,
exists. Let . It is clear that . Assume that . We have
so .
Hence, . We shall show that and We have
That is, is monotonic decreasing and bounded below. Hence, exists.
Let . Assume that . One writes
so . Hence, . Similarly, we have . Also, is a Cauchy sequence. Assume that is not Cauchy. Then,
By using the triangular inequality,
Hence, we have
which gives us
Since and , we get
Observe that . Taking into account that , we get and this contradicts our assumption. Hence, is a Cauchy sequence in . Because is Cauchy, is complete, and is closed; . Now,
Thus, we have converges to . Since
therefore,
Thus,
Theorem 9. Let and be two nonempty closed and convex subsets of a uniformly convex Banach space such that . Suppose is a cyclic Geraghty contraction map. Then, there exists a unique such that . Further, if and , then converges to the best proximity point.
Proof. By Theorem 8, . Suppose such that . Since and were necessarily uniformly convexity of , and . Since , we have and so . Therefore, . Similarly, ; that is, it is a contradiction. Therefore, .
Example 1. Let and be subsets of defined by
Suppose
Here, . For , we have
For and , we have
Then, is a cyclic Geraghty contraction on . Also, .
3. A Best Proximity Point Result for Caristi-Type Mappings
Recently, Du [14] established a direct proof of Caristi’s fixed point theorem without using Zorn’s lemma. In this section, we introduce a generalization of Caristi’s fixed point theorem and provide a proof without using Zorn’s lemma. We start with the following definition.
Definition 10. Let and be nonempty subsets of a metric space . A map is called a semicontraction if for all and, we have
Our related result is as follows.
Theorem 11. Let be a complete metric space and be a pair of nonempty closed subsets of such that is boundedly compact. Also, let a semicontraction map and is a bounded below and lower semicontinuous function . Assume that for all with , there is so that Then, there is so that .
Proof. Assume that for every . Given . Then, . We have . Therefore, there is such that Let us define inductively a sequence , where so that Therefore, Since and is boundedly compact, has a convergent subsequence to . Suppose , . Since , there is so that and . Therefore, We find that . By (30), we get Thus, It is a contradiction, so there is such that .
Definition 12. Let and be nonempty subsets of a metric space . The mapping is a cyclic semicontraction if (i) and (ii)For all and , we have For example, let and be subsets of defined by Suppose , then is cyclic semicontraction on and .
Theorem 13. Let be a pair of nonempty closed subsets of such that is boundedly compact. Suppose that is a cyclic semicontraction map and there exists a bounded below and lower semicontinuous function so that for all . Then, admits a best proximity point.
Proof. The function is proper, so there is with . Put Let . Since , we have . Thus, Hence, and so . Assume that for all . Then, for each , there is so that By Theorem 11, there is so that . Thus, for above , we obtain a contradiction and so has a best proximity point.
From now on, and are nonempty subsets of a Banach space . For , the inward sets of relative to are as follows:
For example, let and be subsets of defined by
Then, for , we have
We now define cyclic weakly inward mappings.
Definition 14. Given . Such is said to be (i)Cyclic inward if for all (ii)Cyclic weakly inward if for all (iii)Cyclic weakly inward contraction if it is cyclic weakly inward and for all , where .
Theorem 15. Let and be closed and convex. Then is cyclic weakly inward if
Proof. Suppose that ((45), (46)) hold. Fix , given . Choose and so that
Thus,
Consequently, . Similarly, for , .
Conversely, assume that is cyclic weakly inward; that is, for all . Given . There exists so that
Since is convex, there is so that
Therefore,
The condition (45) holds. Similarly, the condition (46) is satisfied.
Theorem 16. Let and be closed and convex. Let be a cyclic weakly inward contraction mapping. If is boundedly compact, then has a unique best proximity point in .
Proof. Let denote a Lipschitz constant of . Choose so that . By Theorem 15, (45) is verified. Let with , then there is so that By the definition of a dance, there is so that Hence, and so If for all , , then denote as above by , where is a mapping. Put Note that is continuous and the following holds. Due to Theorem 13, admits a best proximity point, which contradicts (57).
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors read and approved the manuscript.